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Register a new userAsymptotic regimes in elastohydrodynamic and stochastic leveling on a viscous film
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An elastic sheet that deforms near a solid substrate in a viscous fluid is a situation relevant to various dynamical processes in biology, geophysics and engineering. Here, we study the relaxation dynamics of an elastic plate resting on a thin viscous film that is supported by a solid substrate. By combining scaling analysis, numerical simulations and experiments, we identify asymptotic regimes for the elastohydrodynamic leveling of a surface perturbation of the form of a bump, when the flow is driven by either the elastic bending of the plate or thermal fluctuations. In both cases, two distinct regimes are identified when the bump height is either much larger or much smaller than the thickness of the pre-wetted viscous film. Our analysis reveals a distinct crossover between the similarity exponents with the ratio of the perturbation height to the film height.
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We study numerically the effect of thermal fluctuations and of variable fluid-substrate interactions on the spontaneous dewetting of thin liquid films. To this aim, we use a recently developed lattice Boltzmann method for thin liquid film flows, equipped with a properly devised stochastic term. While it is known that thermal fluctuations yield shorter rupture times, we show that this is a general feature of hydrophilic substrates, irrespective of the contact angle. The ratio between deterministic and stochastic rupture times, though, decreases with $theta$. Finally, we discuss the case of fluctuating thin film dewetting on chemically patterned substrates and its dependence on the form of the wettability gradients.
We consider sedimentation of a rigid helical filament in a viscous fluid under gravity. In the Stokes limit, the drag forces and torques on the filament are approximated within the resistive-force theory. We develop an analytic approximation to the exact equations of motion that works well in the limit of a sufficiently large number of turns in the helix (larger than two, typically). For a wide range of initial conditions, our approximation predicts that the centre of the helix itself follows a helical path with the symmetry axis of the trajectory being parallel to the direction of gravity. The radius and the pitch of the trajectory scale as non-trivial powers of the number of turns in the original helix. For the initial conditions corresponding to an almost horizontal orientation of the helix, we predict trajectories that are either attracted towards the horizontal orientation, in which case the helix sediments in a straight line along the direction of gravity, or trajectories that form a helical-like path with many temporal frequencies involved. Our results provide new insight into the sedimentation of chiral objects and might be used to develop new techniques for their spatial separation.
Hydrodynamic interactions between two identical elastic dumbbells settling under gravity in a viscous fluid at low-Reynolds-number are investigated within the point-particle model. Evolution of a benchmark initial configuration is studied, in which the dumbbells are vertical and their centres are aligned horizontally. Rigid dumbbells and pairs of separate beads starting from the same positions tumble periodically while settling down. We find that elasticity (which breaks time-reversal symmetry of the motion) significantly affects the systems dynamics. This is remarkable taking into account that elastic forces are always much smaller than gravity. We observe oscillating motion of the elastic dumbbells, which tumble and change their length non-periodically. Independently of the value of the spring constant, a horizontal hydrodynamic repulsion appears between the dumbbells - their centres of mass move apart from each other horizontally. The shift is fast for moderate values of the spring constant k, and slows down when k tends to zero or to infinity; in these limiting cases we recover the periodic dynamics reported in the literature. For moderate values of the spring constant, and different initial configurations, we observe the existence of a universal time-dependent solution to which the system converges after an initial relaxation phase. The tumbling time and the width of the trajectories in the centre-of-mass frame increase with time. In addition to its fundamental significance, the benchmark solution presented here is important to understand general features of systems with larger number of elastic particles, at regular and random configurations.
A stationary viscous jet falling from an oriented nozzle onto a moving surface is studied, both theoretically and experimentally. We distinguish three flow regimes and classify them by the convexity of the jet shape (concave, vertical and convex). The fluid is modeled as a Newtonian fluid, and the model for the flow includes viscous effects, inertia and gravity. By studying the characteristics of the conservation of momentum for a dynamic jet, the boundary conditions for each flow regime are derived, and the flow regimes are characterized in terms of the process and material parameters. The model is solved by a transformation into an algebraic equation. We make a comparison between the model and experiments, and obtain qualitative agreement.
Marangoni propulsion is a form of locomotion wherein an asymmetric release of surfactant by a body located at the surface of a liquid leads to its directed motion. We present in this paper a mathematical model for Marangoni propulsion in the viscous regime. We consider the case of a thin rigid circular disk placed at the surface of a viscous fluid and whose perimeter has a prescribed concentration of an insoluble surfactant, to which the rest of its surface is impenetrable. Assuming a linearized equation of state between surface tension and surfactant concentration, we derive analytically the surfactant, velocity and pressure fields in the asymptotic limit of low Capillary, Peclet and Reynolds numbers. We then exploit these results to calculate the Marangoni propulsion speed of the disk. Neglecting the stress contribution from Marangoni flows is seen to over-predict the propulsion speed by 50%.
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